# B | Mathematical Phrases, Symbols, and Formulas

Introductory Business StatisticsB | Mathematical Phrases, Symbols, and Formulas

## English Phrases Written Mathematically

When the English says: Interpret this as:
X is at least 4. X ≥ 4
The minimum of X is 4. X ≥ 4
X is no less than 4. X ≥ 4
X is greater than or equal to 4. X ≥ 4
X is at most 4. X ≤ 4
The maximum of X is 4. X ≤ 4
X is no more than 4. X ≤ 4
X is less than or equal to 4. X ≤ 4
X does not exceed 4. X ≤ 4
X is greater than 4. X > 4
X is more than 4. X > 4
X exceeds 4. X > 4
X is less than 4. X < 4
There are fewer X than 4. X < 4
X is 4. X = 4
X is equal to 4. X = 4
X is the same as 4. X = 4
X is not 4. X ≠ 4
X is not equal to 4. X ≠ 4
X is not the same as 4. X ≠ 4
X is different than 4. X ≠ 4
Table B1

## Symbols and Their Meanings

Chapter (1st used) Symbol Spoken Meaning
Sampling and Data The square root of same
Sampling and Data $π π$ Pi 3.14159… (a specific number)
Descriptive Statistics Q1 Quartile one the first quartile
Descriptive Statistics Q2 Quartile two the second quartile
Descriptive Statistics Q3 Quartile three the third quartile
Descriptive Statistics IQR interquartile range Q3Q1 = IQR
Descriptive Statistics $x – x –$ x-bar sample mean
Descriptive Statistics $μμ$ mu population mean
Descriptive Statistics s s sample standard deviation
Descriptive Statistics $s2s2$ s squared sample variance
Descriptive Statistics $σσ$ sigma population standard deviation
Descriptive Statistics $σ2σ2$ sigma squared population variance
Descriptive Statistics $ΣΣ$ capital sigma sum
Probability Topics ${ } {}$ brackets set notation
Probability Topics $SS$ S sample space
Probability Topics $AA$ Event A event A
Probability Topics $P ( A ) P ( A )$ probability of A probability of A occurring
Probability Topics $P(A|B) P(A|B)$ probability of A given B prob. of A occurring given B has occurred
Probability Topics $P(A∪B) P(A∪B)$ prob. of A or B prob. of A or B or both occurring
Probability Topics $P(A∩B) P(A∩B)$ prob. of A and B prob. of both A and B occurring (same time)
Probability Topics A A-prime, complement of A complement of A, not A
Probability Topics P(A') prob. of complement of A same
Probability Topics G1 green on first pick same
Probability Topics P(G1) prob. of green on first pick same
Discrete Random Variables PDF prob. density function same
Discrete Random Variables X X the random variable X
Discrete Random Variables X ~ the distribution of X same
Discrete Random Variables $≥ ≥$ greater than or equal to same
Discrete Random Variables $≤ ≤$ less than or equal to same
Discrete Random Variables = equal to same
Discrete Random Variables not equal to same
Continuous Random Variables f(x) f of x function of x
Continuous Random Variables pdf prob. density function same
Continuous Random Variables U uniform distribution same
Continuous Random Variables Exp exponential distribution same
Continuous Random Variables f(x) = f of x equals same
Continuous Random Variables m m decay rate (for exp. dist.)
The Normal Distribution N normal distribution same
The Normal Distribution z z-score same
The Normal Distribution Z standard normal dist. same
The Central Limit Theorem $X – X –$ X-bar the random variable X-bar
The Central Limit Theorem $μ x – μ x –$ mean of X-bars the average of X-bars
The Central Limit Theorem $σ x – σ x –$ standard deviation of X-bars same
Confidence Intervals CL confidence level same
Confidence Intervals CI confidence interval same
Confidence Intervals EBM error bound for a mean same
Confidence Intervals EBP error bound for a proportion same
Confidence Intervals t Student's t-distribution same
Confidence Intervals df degrees of freedom same
Confidence Intervals $t α 2 t α 2$ student t with α/2 area in right tail same
Confidence Intervals $p′ p′$ p-prime sample proportion of success
Confidence Intervals $q′ q′$ q-prime sample proportion of failure
Hypothesis Testing $H 0 H 0$ H-naught, H-sub 0 null hypothesis
Hypothesis Testing $H a H a$ H-a, H-sub a alternate hypothesis
Hypothesis Testing $H 1 H 1$ H-1, H-sub 1 alternate hypothesis
Hypothesis Testing $αα$ alpha probability of Type I error
Hypothesis Testing $β β$ beta probability of Type II error
Hypothesis Testing $X1 – – X2 ¯ X1 – – X2 ¯$ X1-bar minus X2-bar difference in sample means
Hypothesis Testing $μ 1 − μ 2 μ 1 − μ 2$ mu-1 minus mu-2 difference in population means
Hypothesis Testing $P ′ 1 − P ′ 2 P ′ 1 − P ′ 2$ P1-prime minus P2-prime difference in sample proportions
Hypothesis Testing $p 1 − p 2 p 1 − p 2$ p1 minus p2 difference in population proportions
Chi-Square Distribution $Χ 2 Χ 2$ Ky-square Chi-square
Chi-Square Distribution $OO$ Observed Observed frequency
Chi-Square Distribution $EE$ Expected Expected frequency
Linear Regression and Correlation y = a + bx y equals a plus b-x equation of a straight line
Linear Regression and Correlation $y^y^$ y-hat estimated value of y
Linear Regression and Correlation $rr$ sample correlation coefficient same
Linear Regression and Correlation $εε$ error term for a regression line same
Linear Regression and Correlation SSE Sum of Squared Errors same
F-Distribution and ANOVA F F-ratio F-ratio
Table B2 Symbols and their Meanings

## Formulas

 Symbols you must know Population Sample $NN$ Size $nn$ $μμ$ Mean $x_x_$ $σ2σ2$ Variance $s2s2$ $σσ$ Standard deviation $ss$ $pp$ Proportion $p′p′$ Single data set formulae Population Sample $μ = E(x) = 1N ∑ i=1 N (xi) μ=E(x)=1N∑ i=1 N (xi)$ Arithmetic mean $x – = 1n ∑ i=1 n (xi) x – =1n∑ i=1 n (xi)$ Geometric mean $x~ = (∏i=1nXi) 1nx~= (∏i=1nXi) 1n$ $Q3=3(n+1)4Q3=3(n+1)4$, $Q1=(n+1)4Q1=(n+1)4$ Inter-quartile range$IQR=Q3−Q1IQR=Q3−Q1$ $Q3=3(n+1)4Q3=3(n+1)4$, $Q1=(n+1)4Q1=(n+1)4$ $σ2=1N∑ i=1 N (xi−μ)2σ2=1N∑ i=1 N (xi−μ)2$ Variance $s2=1n∑ i=1 n (xi−x_)2s2=1n∑ i=1 n (xi−x_)2$ Single data set formulae Population Sample $μ = E(x) = 1N ∑ i=1 N (mi·fi) μ=E(x)=1N∑ i=1 N (mi·fi)$ Arithmetic mean $x – = 1n ∑ i=1 n (mi·fi) x – =1n∑ i=1 n (mi·fi)$ Geometric mean $x~ = (∏i=1nXi) 1nx~= (∏i=1nXi) 1n$ $σ2=1N∑ i=1 N (mi−μ)2 ·fiσ2=1N∑ i=1 N (mi−μ)2·fi$ Variance $s2=1n∑ i=1 n (mi−x_)2·fis2=1n∑ i=1 n (mi−x_)2·fi$ $CV=σμ·100CV=σμ·100$ Coefficient of variation $CV=sx_·100CV=sx_·100$
Table B3
 Basic probability rules $P(A∩B)=P(A|B)·P(B)P(A∩B)=P(A|B)·P(B)$ Multiplication rule $P(A∪B)=P(A)+P(B)−P(A∩B)P(A∪B)=P(A)+P(B)−P(A∩B)$ Addition rule $P(A∩B)=P(A)·P(B)P(A∩B)=P(A)·P(B)$ or $P(A|B)=P(A)P(A|B)=P(A)$ Independence test Hypergeometric distribution formulae $nCx=(nx)=n!x!(n−x)!nCx=(nx)=n!x!(n−x)!$ Combinatorial equation $P(x)=(Ax)(N−An−x)(Nn)P(x)=(Ax)(N−An−x)(Nn)$ Probability equation $E(X)=μ=npE(X)=μ=np$ Mean $σ2=(N−nN−1)np(q)σ2=(N−nN−1)np(q)$ Variance Binomial distribution formulae $P(x)=n!x!(n−x)!px(q)n−xP(x)=n!x!(n−x)!px(q)n−x$ Probability density function $E(X)=μ=npE(X)=μ=np$ Arithmetic mean $σ2=np(q)σ2=np(q)$ Variance Geometric distribution formulae $P(X=x)=(1−p)x−1(p)P(X=x)=(1−p)x−1(p)$ Probability when $xx$ is the first success. Probability when $xx$ is the number of failures before first success $P(X=x)=(1−p)x(p)P(X=x)=(1−p)x(p)$ $μ=1pμ=1p$ Mean Mean $μ=1−ppμ=1−pp$ $σ2=(1−p)p2σ2=(1−p)p2$ Variance Variance $σ2=(1−p)p2σ2=(1−p)p2$ Poisson distribution formulae $P(x)=e−μμxx!P(x)=e−μμxx!$ Probability equation $E(X)=μE(X)=μ$ Mean $σ2=μσ2=μ$ Variance Uniform distribution formulae $f(x)=1b−af(x)=1b−a$ for $a≤x≤ba≤x≤b$ PDF $E(X)=μ=a+b2E(X)=μ=a+b2$ Mean $σ2=(b−a)212σ2=(b−a)212$ Variance Exponential distribution formulae $P(X≤x)=1−e−mxP(X≤x)=1−e−mx$ Cumulative probability $E(X)=μ=1mE(X)=μ=1m$ or $m=1μm=1μ$ Mean and decay factor $σ2=1m2=μ2σ2=1m2=μ2$ Variance
Table B4
 The following page of formulae requires the use of the "$ZZ$", "$tt$", "$χ2χ2$" or "$FF$" tables. $Z=x−μσZ=x−μσ$ Z-transformation for normal distribution $Z=x−np′np′(q′)Z=x−np′np′(q′)$ Normal approximation to the binomial Probability (ignores subscripts)Hypothesis testing Confidence intervals[bracketed symbols equal margin of error](subscripts denote locations on respective distribution tables) $Zc=x¯-μ0σnZc=x¯-μ0σn$ Interval for the population mean when sigma is known$x¯±[Z(α/2)σn]x¯±[Z(α/2)σn]$ $Zc=x¯-μ0snZc=x¯-μ0sn$ Interval for the population mean when sigma is unknown but $n>30n>30$$x¯±[Z(α/2)sn]x¯±[Z(α/2)sn]$ $tc=x¯-μ0sntc=x¯-μ0sn$ Interval for the population mean when sigma is unknown but $n<30n<30$$x¯±[t(n−1),(α/2)sn]x¯±[t(n−1),(α/2)sn]$ $Zc=p′-p0p0q0nZc=p′-p0p0q0n$ Interval for the population proportion$p′±[Z(α/2)p′q′n]p′±[Z(α/2)p′q′n]$ $tc= d–-δ0 sd n tc= d–-δ0 sd n$ Interval for difference between two means with matched pairs$d–±[t(n−1),(α/2) sdn ]d–±[t(n−1),(α/2)sdn]$ where $sdsd$ is the deviation of the differences $Zc=(x1–-x2–)−δ0σ12n1+σ22n2Zc=(x1–-x2–)−δ0σ12n1+σ22n2$ Interval for difference between two means when sigmas are known$(x1–-x2–)±[Z(α/2)σ12n1+σ22n2](x1–-x2–)±[Z(α/2)σ12n1+σ22n2]$ $tc=(x¯1-x¯2)-δ0((s1)2n1+(s2)2n2)tc=(x¯1-x¯2)-δ0((s1)2n1+(s2)2n2)$ Interval for difference between two means with equal variances when sigmas are unknown$(x¯1-x¯2)±[tdf,(α/2)((s1)2n1+(s2)2n2)](x¯1-x¯2)±[tdf,(α/2)((s1)2n1+(s2)2n2)]$ where $df= ( (s1)2n1 + (s2)2n2 )2 (1n1−1) ((s1)2n1) + (1n2−1) ((s2)2n2) df= ( (s1)2n1 + (s2)2n2 )2 (1n1−1) ((s1)2n1) + (1n2−1) ((s2)2n2)$ $Zc=(p′1-p′2)−δ0p′1(q′1)n1+p′2(q′2)n2Zc=(p′1-p′2)−δ0p′1(q′1)n1+p′2(q′2)n2$ Interval for difference between two population proportions$(p′1-p′2)±[Z(α/2)p′1(q′1)n1+p′2(q′2)n2 ](p′1-p′2)±[Z(α/2)p′1(q′1)n1+p′2(q′2)n2]$ $χc2=(n−1)s2σ02χc2=(n−1)s2σ02$ Tests for GOF, Independence, and Homogeneity$χc2=Σ(O−E)2Eχc2=Σ(O−E)2E$where O = observed values and E = expected values $Fc=s12s22Fc=s12s22$ Where $s12s12$ is the sample variance which is the larger of the two sample variances The next 3 formulae are for determining sample size with confidence intervals.(note: E represents the margin of error) $n=Z(a2)2σ2E2n=Z(a2)2σ2E2$ Use when sigma is known $E=x¯−μE=x¯−μ$ $n=Z(a2)2(0.25)E2n=Z(a2)2(0.25)E2$ Use when $p′p′$ is unknown $E=p′−pE=p′−p$ $n=Z(a2)2[p′(q′)]E2n=Z(a2)2[p′(q′)]E2$ Use when $p′p′$ is unknown $E=p′−pE=p′−p$
Table B5
 Simple linear regression formulae for $y=a+b(x)y=a+b(x)$ $r=Σ[(x−x¯)(y−y¯)]Σ(x−x¯)2*Σ(y−y¯)2=SxySxSy=SSRSSTr=Σ[(x−x¯)(y−y¯)]Σ(x−x¯)2*Σ(y−y¯)2=SxySxSy=SSRSST$ Correlation coefficient $b=Σ[(x−x¯)(y−y¯)]Σ(x−x¯)2=SxySSx=ry,x(sysx)b=Σ[(x−x¯)(y−y¯)]Σ(x−x¯)2=SxySSx=ry,x(sysx)$ Coefficient b (slope) $a=y¯−b(x¯)a=y¯−b(x¯)$ y-intercept $se2=Σ(yi−y^i)2 n−k=Σi=1nei2n−kse2=Σ(yi−y^i)2 n−k=Σi=1nei2n−k$ Estimate of the error variance $Sb=se2(xi−x¯)2=se2(n−1)sx2Sb=se2(xi−x¯)2=se2(n−1)sx2$ Standard error for coefficient b $tc=b−β0sbtc=b−β0sb$ Hypothesis test for coefficient β $b±[tn−2,α/2Sb]b±[tn−2,α/2Sb]$ Interval for coefficient β $y^±[tα/2*se(1n+(xp−x¯)2sx)]y^±[tα/2*se(1n+(xp−x¯)2sx)]$ Interval for expected value of y $y^±[tα/2*se(1+1n+(xp−x¯)2sx)]y^±[tα/2*se(1+1n+(xp−x¯)2sx)]$ Prediction interval for an individual y ANOVA formulae $SSR=Σi=1n(y^i−y¯)2SSR=Σi=1n(y^i−y¯)2$ Sum of squares regression $SSE=Σi=1n(y^i−y¯i)2SSE=Σi=1n(y^i−y¯i)2$ Sum of squares error $SST=Σi=1n(yi−y¯)2SST=Σi=1n(yi−y¯)2$ Sum of squares total $R2=SSRSSTR2=SSRSST$ Coefficient of determination
Table B6
 The following is the breakdown of a one-way ANOVA table for linear regression. Source of variation Sum of squares Degrees of freedom Mean squares F-ratio Regression $SSRSSR$ $11$ or $k−1k−1$ $MSR=SSRdfRMSR=SSRdfR$ $F=MSRMSEF=MSRMSE$ Error $SSESSE$ $n−kn−k$ $MSE=SSEdfEMSE=SSEdfE$ Total $SSTSST$ $n−1n−1$
Table B7 Do you know how you learn best?
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